Modified Projection Algorithms for Solving the Split Equality Problems

Abstract

The split equality problem (SEP) has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Byrne and Moudafi (2013) proposed a CQ algorithm for solving it. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively and performs an additional projection step onto two half-spaces in each iteration. We further propose a relaxation scheme for the self-adaptive projection algorithm by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Weak convergence results for both algorithms are analyzed.

1. Introduction

The split equality problem (SEP) was introduced by Moudafi [1] and its interest covers many situations, for instance, in domain decomposition for PDE’s, game theory, and intensity-modulated radiation therapy (IMRT) (see [2–7] for more details). Let , , and be real Hilbert spaces; let and be two nonempty closed convex sets; let and be two bounded linear operators. The SEP can mathematically be formulated as the problem of finding and with the property which allows asymmetric and partial relations between the variables and . If and , then the split equality problem (1) reduces to the split feasibility problem (originally introduced in Censor and Elfving [8]) which is to find with .

For solving the SEP (1), Moudafi [1] introduced the following alternating algorithm: where and and are the spectral radii of and , respectively. By studying the projected Landweber algorithm of the SEP (1) in a product space, Byrne and Moudafi [7] obtained the following algorithm: where , the stepsize at the iteration , is chosen in the interval. It is easy to see that the alternating algorithm (2) is sequential but the algorithm (3) is simultaneous.

Observe that in the algorithms (2) and (3), the determination of the stepsize depends on the operator (matrix) norms and (or the largest eigenvalues of and ). This means that, in order to implement the alternating algorithm (2), one has first to compute (or, at least, estimate) operator norms of and , which is in general not an easy work in practice. Considering this, Dong and He [9] proposed algorithms without prior knowledge of operator norms.

In this paper, we first propose a modification for algorithm (3), inspired by Tseng [10] (also see [11]). Our modified projection method computes the stepsize adaptively and performs an additional projection step onto two half-spaces, and , in each iteration. Then we give a relaxation scheme for this modification by replacing the orthogonal projections onto the sets and by projections onto the two half-spaces and , respectively. Since projections onto half-spaces can be directly calculated, the relaxed scheme will be more practical and easily implemented.

The rest of this paper is organized as follows. In the next section, some useful facts and tools are given. The weak theorem of the proposed self-adaptive projection algorithm is obtained in Section 3. In Section 4, we consider a relaxed self-adaptive projection algorithm, where the sets and are level sets of convex functions.

2. Preliminaries

In this section, we review some definitions and lemmas which will be used in this paper.

Let be a Hilbert space and let be the identity operator on . If is a differentiable functional, then denote by the gradient of . If is a subdifferentiable functional, then denote by the subdifferential of . Given a sequence in , stands for the set of cluster points in the weak topology. “” (resp., “”) means the strong (resp., weak) convergence of to .

Definition 1. A sequence is said to be asymptotically regular if

Definition 2. The graph of an operator is called to be weakly-strongly closed if with strongly converging to and weakly converging to ; then .

The next lemma is well known (see [10, 12]) and shows that the maximal monotone operators are weakly-strongly closed.

Lemma 3. Let be a Hilbert space and let be a maximal monotone mapping. If is a sequence in bounded in norm and converging weakly to some and is a sequence in converging strongly to some and for all , then .

The projection is an important tool for our work in this paper. Let be a closed convex subset of real Hilbert space . Recall that the (nearest point or metric) projection from onto , denoted by, is defined in such a way that, for each , is the unique point in such that

The following two lemmas are useful characterizations of projections.

Lemma 4. Given and , then if and only if

Lemma 5. For any and , it holds (i); (ii).

Throughout this paper, assume that the split equality problem (1) is consistent and denote by the solution of (1); that is, Then is closed, convex, and nonempty. The split equality problem (1) can be written as the following minimization problem: where is an indicator function of the set defined by By writing down the optimality conditions, we obtain which implies, for and , which in turn leads to the fixed point formulation Since and , we have The following proposition shows that solutions of the fixed point equations (17) are exactly the solutions of the SEP (1).

Proposition 6 (see [9]). Given and , then solves the SEP (1) if and only if solves the fixed point equations (13).

3. A Self-Adaptive Projection Algorithm

Based on Proposition 6, we construct a self-adaptive projection algorithm for the fixed point equations (13) and prove the weak convergence of the proposed algorithm.

Define the function by and the function by The self-adaptive projection algorithm is defined as follows.

Algorithm 7. Given constants ,  ,   and , let and be arbitrary. For , compute where is chosen to be the largest satisfying Construct the half-spaces and , the bounding hyperplanes of which support and at and , respectively, Set If then set ; otherwise, set .

In this algorithm, (19) involves projection onto half-spaces (resp., ) rather than onto the set (resp., ) and it is obvious that projections on (resp., ) are very simple. It is easy to show and . The last step is used to reduce the inner iterations for searching the stepsize .

Lemma 8. The search rule (17) is well defined. Besides , where

Proof. Obviously, . If , then this lemma is proved; otherwise, if , by the search rule (17), we know that must violate inequality (17); that is, On the other hand, we have Consequently, we get which completes the proof.

Theorem 9. Let be the sequence generated by Algorithm 7 and let and be nonempty closed convex sets in and with simple structures, respectively. If is nonempty, then converges weakly to a solution of the SEP (1).

Proof. Let ; that is, , , and . Define ; then we have where the first inequality follows from nonexpansivity of the projection mapping . Similarly, defining , we get Adding the above inequalities, we obtain where the equality follows from and , the second inequality follows from (17), and the last follows from (16) and Lemma 3 and ,. Using the fact and , we have which with (27) implies that Consequently, the sequence is decreasing and lower bounded by 0 and thus converges to some finite limit, say, . Moreover, and are bounded. This implies that From (30), we get
Let ; then there exist the two subsequences and of and which converge weakly to and , respectively. We will show that is a solution of the SEP (1). The weak convergence of to and lower semicontinuity of the squared norm imply that that is, .
By noting that the two equalities in (16) can be rewritten as and that the graphs of the maximal monotone operators, and , are weakly-strongly closed and by passing to the limit in the last inclusions, we obtain, from (30), that Hence .
To show the uniqueness of the weak cluster points, we will use the same strick as in the celebrated Opial Lemma. Indeed, let be other weak cluster point of . By passing to the limit in the relation we obtain Reversing the role of and , we also have By adding the two last equalities, we obtain Hence ; this implies that the whole sequence weakly converges to a solution of the SEP (1), which completes the proof.

4. A Relaxed Self-Adaptive Projection Algorithm

In Algorithm 7, we must calculate the orthogonal projections, and , many times even in one iteration step, so they should be assumed to be easily calculated; however, sometimes it is difficult or even impossible to compute them. In this case, we always turn to relaxed methods [13, 14], which were introduced by Fukushima [15] and are more easily implemented. For solving the SEP (1), Moudafi [16] followed the ideas of Fukushima [15] and introduced a relaxed alternating algorithm which depends on the norms and . In this section, we propose a relaxed scheme for the self-adaptive Algorithm 7.

Assume that the convex sets and are given by where and are convex functions which are subdifferentiable on and , respectively, and we assume that their subdifferentials are bounded on bounded sets.

In the th iteration, let and be two sequences of closed convex sets defined by where and where .

It is easy to see that and for every .

Algorithm 10. Given constants ,  ,  ,  and , let and be arbitrary. For , compute where is chosen to be the largest satisfying Construct the half-spaces and the bounding hyperplanes of which support and at and , respectively, Set If then set ; otherwise, set .

Following the proof of Lemma 8, we easily obtain the following.

Lemma 11. The search rule (43) is well defined. Besides , where

Theorem 12. Let be the sequence generated by Algorithm 10 and let and be nonempty closed convex sets in and with simple structures, respectively. If is nonempty, then converges weakly to a solution of the SEP (1).

Proof. Let ; that is, , , and . Following the similar proof of Theorem 9, we obtain Let . Then the sequence is decreasing and lower bounded by 0 for that and thus converges to some finite limit, say, . Moreover, and are bounded. This implies that Therefore, we have
Next we show that the sequence generated by Algorithm 10 weakly converges to a solution of the SEP (1). Let ; then there exist the two subsequences and of and which converge weakly to and , respectively. The weak convergence of to and the lower semicontinuity of the squared norm imply that that is, .
Since , we have Thus where satisfies for all . The lower semicontinuity of and the first formula of (49) lead to and therefore .
Likewise, since , we have Thus where satisfies for all . Again, the lower semicontinuity of and the second formula of (49) lead to and therefore . Hence .
Following the same argument of Theorem 9, we can show the uniqueness of the weak cluster points and hence the whole sequence weakly converges to a solution of the SEP (1), which completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their thanks to Abdellatif Moudafi for helpful correspondence. The work was supported by National Natural Science Foundation of China (no. 11201476) and Fundamental Research Funds for the Central Universities (no. 3122013D017).